For the sake of brevity we will take the special case of a wave travelling parallel to the magnetic force in the direction of the axis of z. Then supposing that all the ions are of the same kind, and that there are _n_ of these each with mass _m_ and charge _e_ per unit volume, the equations representing the field are (see ELECTRIC WAVES):-- dX0 d[xi] d[beta] K0 --- + 4[pi]ne ----- = -------; dt dt dz dX[0] d[beta] ----- = -------; dz dt dY0 d[eta] d[alpha] K0 --- + 4[pi]ne ------ = - -------- dt dt dz dY0 d[alpha] --- = - --------; dz dt d²[xi] d[xi] / 4[pi] \ d[eta] m ------ + R1 ----- + a[xi] = ( X0 + ----- ne[xi] ) e + He ------ dt² dt \ 3 / dt d²[eta] d[eta] / 4[pi] \ d[xi] m ------- + R1 ------ + a[eta] = ( Y0 + ----- ne[eta] ) e - He -----; dt² dt \ 3 / dt where H is the external magnetic field, X0, Y0 the components of the part of the electric force in the wave not due to the charges on the atoms, [alpha] and [beta] the components of the magnetic force, [xi] and [eta] the co-ordinates of an ion, R1 the coefficient of resistance to the motion of the ions, and [alpha] the force at unit distance tending to bring the ion back to its position of equilibrium, K0 the specific inductive capacity of a vacuum. If the variables are proportional to [epsilon]^[l(pt - qz)] we find by substitution that q is given by the equation 4[pi]ne²p²P 4[pi]ne³Hp³ q² - K0p² - ----------- = ± -----------, P² - H²e²p² P² - H²e²p² where P = (a - (4/3)[pi]ne²) + R1[iota]p - mp², or, by neglecting R, P = m(s² - p²), where s is the period of the free ions. If, q1², q2² are the roots of this equation, then corresponding to q1 we have X0 = [iota]Y0 and to q2 X0 = -[iota]Y0. We thus get two oppositely circular-polarized rays travelling with the velocities p/q1 and p/q2 respectively. Hence if v1, v2 are these velocities, and v the velocity when there is no magnetic field, we obtain, if we neglect terms in H², 1 1 4[pi]ne³Hp --- = -- + ------------, v1² v² m²(s² - p²)² 1 1 4[pi]ne³Hp --- = -- - ------------. v2² v² m²(s² - p²)² The rotation r of the plane of polarization per unit length / 1 1 \ 2[pi]ne³Hp²v = ½p ( --- - --- ) = -------------. \ v1 v2 / m²(s² - p²)² Since 1/v² = K0 + 4[pi]ne²/m(s² - p²), we have if µ is the refractive index for light of frequency p, and v0 the velocity of light in vacuo. µ² - 1 = 4[pi]ne²v²0 / m(s² - p²) (1) So that we may put r = (µ² - 1)²p²H / s[pi]µne v0³ (2)